Global dynamics of a plant-herbivore model with toxin-determined functional response

Studies of the dynamics of plant-herbivore intera ctions that explicitly address model robustness are important in assessing uncertainty. Hence, identifying conditions that guarantee the global stability of plant-herbivore systems can be used to assess the rationale involved in, for example, the selection of management and/or control decisions. The model used here to illustrate these issues is naturally capable of supporting complex dynamics; the result of the explicit incorporation of plant toxicity in the functional response. Unlike the traditional Holling Type 2 functional response, the selected toxin-determined functional response loses its monotonicity at high levels of plant toxicity. Systems with nonmonotone functional responses are capable of supporting multiple interior equilibria and bistable attractors. Therefore, identifying conditions that guarantee global stability is not only mathematically challenging but important to scientists. We are able to find necessary and sufficient condition on the nonexistence of a closed orbit via the transformation of the model to a new equivalent system. The Poincaré-Bendixson theorem is used to show that the existence of a unique interior equilibrium point guarantees its global asymptotical stability whenever it is locally asymptotically stable. It is shown that, whenever there are multiple interior equilibria, the local stability of the "first interior equilibrium" implies model bistability. In other words, the phase space is characterized by two subregions: the basins of attraction of two stable equilibria, the interior and the boundary equilibria.